U.S. patent number 4,258,322 [Application Number 05/903,800] was granted by the patent office on 1981-03-24 for electromagnetic subsoil prospecting process using an asymptotic low frequency range.
This patent grant is currently assigned to Compagnie Generale de Geophysique. Invention is credited to Andre Cecchini, Jean-Pierre Rocroi.
United States Patent |
4,258,322 |
Rocroi , et al. |
March 24, 1981 |
Electromagnetic subsoil prospecting process using an asymptotic low
frequency range
Abstract
In a sub-soil prospecting process a vertical magnetic field
transmitter dipole and a receiver dipole are placed on the ground,
away from each other. The receiver dipole is oriented to detect a
radial horizontal magnetic field relative to the transmitter
dipole. At low frequencies, the magnetic field detected assumes
asymptotic values, and by measuring these values an apparent
alternating current resistivity of the sub-soil is deduced. By
varying the transmitter-receiver distance, and by simultaneously
using a direct current electric prospecting process, a sub-soil
model is determined which is less ambiguous than those supplied by
electric or induction prospecting alone.
Inventors: |
Rocroi; Jean-Pierre (Massy,
FR), Cecchini; Andre (Chatenay-Malabry,
FR) |
Assignee: |
Compagnie Generale de
Geophysique (Massy, FR)
|
Family
ID: |
9190544 |
Appl.
No.: |
05/903,800 |
Filed: |
May 8, 1978 |
Foreign Application Priority Data
|
|
|
|
|
May 9, 1977 [FR] |
|
|
77 14085 |
|
Current U.S.
Class: |
324/335;
324/345 |
Current CPC
Class: |
G01V
3/108 (20130101) |
Current International
Class: |
G01V
3/10 (20060101); G01V 003/10 () |
Field of
Search: |
;324/6,334,335,345 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Enenshtein, B. S., "Interpretation Two-Layer Curves . . . for
P.sub.2 <P.sub.1 ", Bull. Acad. of Sci., USSR; Geoph. Series;
No.9, (Translation pp. 733-736). .
Duprat et al., "Exemples . . . L'Interpetation de Sondages
Electriques",Geophysical Prospecting, vol. 21, No. 3, Sep. 1973,
pp. 543-559. .
Kunetz et al., "Traitement . . . of Electrical Soundings",
Geophysical Prospecting, vol. XVIII, No. 2, Jun. 1970, pp. 157-197.
.
Rocroi, J. P., "Contribution . . . en Prospection Electrique",
Geophysical Prospecting, vol. 18, No. 4, 1975, pp.
765-778..
|
Primary Examiner: Strecker; Gerard R.
Attorney, Agent or Firm: Schwartz, Jeffery, Schwaab, Mack,
Blumenthal & Koch
Claims
We claim:
1. A sub-soil prospecting method, comprising the steps of: (a)
placing above ground a magnetic transmitter dipole and a magnetic
receiver dipole spaced at a distance from each other in a
prospecting area;
(b) energizing the transmitter dipole with alternating current of
frequency below a predetermined low frequency;
(c) sensing at an output of the receiver dipole a signal
representing a detected magnetic field, said predetermined low
frequency being a maximum frequency of an asymptotic range in which
the magnetic field represented by the receiver dipole output signal
is linearly related to the energizing frequency; and
determining a quantity related to the ratio of the magnitude of
said sensed signal to the value of said energizing frequency.
2. A sub-soil prospecting method according to claim 1, further
including the steps of varying the distance between the transmitter
dipole and the receiver dipole, and repeating steps (b) and
(c).
3. A subsoil prospecting method according to claim 1 or claim 2,
wherein said predetermined low frequency is no greater than about
100 Hz.
4. A subsoil prospecting method according to claim 1 or claim 2,
wherein said step of energizing the transmitter dipole includes the
step of varying the frequency of said alternating current, and
wherein the step of sensing a signal at the receiver dipole output
includes sensing a signal for each transmitter dipole energizing
frequency.
5. A subsoil prospecting method according to claim 4, wherein the
frequency of the alternating current is varied by stepwise
reduction of the frequency.
6. A subsoil prospecting method according to claim 5, wherein the
frequency reduction is conducted in successive frequency steps
related by a geometrical progression to give equal steps on a
logarithmic scale.
7. A subsoil prospecting method according to claim 1 or claim 2,
wherein step (a) includes the steps of orientating the transmitter
dipole as a vertical magnetic dipole, and orientating the receiving
dipole with its axis directed towards the transmitter dipole to
detect a substantially radial horizontal magnetic field with
respect to the transmitter dipole.
8. A sub-soil prospecting method according to claim 7, further
comprising the steps of placing a second receiver dipole at
substantially the same location as said first-mentioned receiver
dipole, and orientating the second receiver dipole to detect a
substantially vertical magnetic field, whereby the asymptotic
values of the relation between the detected vertical magnetic field
and the detected horizontal magnetic field may be determined as a
function of the distance between the transmitter dipole and the
receiver dipoles.
9. A sub-soil prospecting method according to claim 1 or claim 2,
wherein the magnetic transmitter and receiver dipoles are placed
substantially at ground level.
10. A subsoil prospecting method according to claim 2, wherein the
distance between the transmitter and receiver dipoles is
progressively increased.
11. A sub-soil prospecting method according to claim 2, wherein the
distance between the transmitter and receiver dipoles is varied in
geometrical progression for each repetition of steps (b) and (c) to
give equal steps on a logarithmic scale.
12. A sub-soil prospecting method according to claim 2, further
comprising the steps of constructing from the signals representing
detected magnetic fields an alternating-current apparent
resistivity curve as a function of the distance between the
transmitter and receiver dipoles.
13. A sub-soil prospecting method according to claim 12, further
comprising the step of carrying out at the same prospecting area a
direct-current electric prospecting method comprising the steps
of:
placing in the ground two exciter electrodes at a distance from
each other;
placing two closely spaced detector electrodes in the ground
between the exciter electrodes;
applying a direct-current potential across the exciter
electrodes;
measuring the resultant potential between the detector
electrodes;
repeating said potential measurement for different distances
between the exciter electrodes; and
constructing a direct-current apparent sub-soil resistivity curve
as a function of the distance between the exciter electrodes.
14. A sub-soil prospecting method according to claim 13, further
comprising the step of combining the measurements taken using said
direct-current prospecting method with the receiver dipole output
signals of the alternating-current prospecting method to construct
a relatively less ambiquous model of the sub-soil.
15. A sub-soil prospecting method according to claim 14, wherein
said step of combining includes the steps of:
determining an initial model of the subsoil;
determining theoretical direct-current and alternating-current
apparent resistivity curves corresponding to the initial model;
adjusting the theoretical curves to the constructed curves using
the least squares successive approximations techniques; and
adjusting the initial model to agree with the adjusted theoretical
curves.
16. A sub-soil prospecting method, comprising the steps of:
(a) placing a magnetic transmitter dipole and a magnetic receiver
dipole at respective locations above ground spaced at a distance
from each other in a prospecting area;
(b) energizing the magnetic transmitter dipole with a low frequency
alternating current;
(c) sensing at an output of the magnetic receiver dipole a signal
representing a detected magnetic field; and
(d) repeating steps (b) and (c) while varying the frequency of said
alternating current until reaching a low-frequency asymptotic range
in which the detected magnetic field represented by the receiver
dipole output signal is substantially linearly related to the
alternating current energizing frequency; and
for each repetition of steps (b) and (c), determining a quantity
related to the ratio of the magnitude of said sensed signal to the
value of said energizing frequency.
17. A sub-soil prospecting method according to claim 16, wherein
step (d) comprises repeating steps (b) and (c) for successively
decreasing low frequencies until the output signal of the receiver
dipole lies within said asymptotic range.
18. A sub-soil prospecting method according to claim 17, wherein
the frequency of said alternating current is successively reduced
in steps.
19. A sub-soil prospecting method according to claim 18, wherein
the successive steps of frequency reduction are related by
geometrical progression to give equal steps on a logarithmic
scale.
20. A sub-soil prospecting method according to claim 16, wherein
step (a) includes the steps of orientating the transmitter dipole
as a vertical magnetic dipole and orientating the receiver dipole
with its axis directed substantially toward the transmitter dipole
to detect a substantially radial horizontal magnetic field with
respect to the transmitter dipole.
21. A sub-soil prospecting method according to claim 20, wherein
both said transmitter and receiver dipoles are placed substantially
at ground level.
22. A sub-soil prospecting method according to claim 21, further
comprising the step of determining the ratio of the value of the
detected radial horizontal magnetic field represented by the
receiver dipole output signal to the value of the energizing
alternating current frequency, said radio being substantially
constant when the energizing alternating frequency is within said
low-frequency asymptotic range.
23. A sub-soil prospecting method according to claim 20, further
comprising placing a second receiver dipole at substantially the
same location as said first-mentioned receiver dipole, orientating
the second receiver dipole to detect a substantially vertical
magnetic field, sensing at an output of the second magnetic
receiver dipole a signal representing the detected vertical
magnetic field, and determining a quantity related to the ratio of
the value of the detected horizontal radial magnetic field to the
value of the detected vertical magnetic field.
24. A sub-soil prospecting method according to claim 16, wherein
step (d) further comprises repeating steps (b) and (c) while
varying the distance between the transmitter and receiver dipoles
whereby at least one receiver dipole output signal representing a
detected magnetic field value lying within the low-frequency
asymptotic range is obtained for each transmitter dipole to
receiver dipole distance.
25. A sub-soil prospecting method according to claim 24, wherein
the distance between the transmitter and receiver dipoles is
successively increased, step (d) comprises repeating steps (b) and
(c) once for each transmitter dipole to receiver dipole distance,
and the frequency of the energizing alternating current is within
the low-frequency asymptotic range for the initial transmitter
dipole to receiver dipole distance.
26. A sub-soil prospecting method according to claim 25, wherein
the distance between the transmitter dipole and the receiver dipole
is successively increased in steps, the increasing distances
forming a geometrical progression to give equal steps on a
logarithmic scale.
27. A sub-soil prospecting method according to claim 24, further
comprising the step of recording a receiver dipole output signal
value representing a detected magnetic field lying within the
asymptotic range as a function of the transmitter dipole to
receiver dipole distance.
28. A sub-soil prospecting method according to claim 27, further
comprising the step of determining an alternating current apparent
resistivity value from each said at least one magnetic receiver
dipole output signal, the alternating current apparent resistivity
being a function of transmitter dipole to receiver dipole distance
and being substantially independent of the alternating current
energizing frequency.
29. A sub-soil prospecting method according to claim 28, wherein
the magnetic transmitter dipole is placed having a dipole axis
thereof arranged vertically, the magnetic receiver dipole is placed
having a dipole axis thereof arranged horizontally and directed
substantially radially towards the transmitter dipole, and the
alternating current apparent resistivity values are determined in
accordance with the relationship: ##EQU6## wherein .rho..sub.a is
the alternating current apparent resistivity in ohms-meter;
e is the efficiency coefficient of the transmitter dipole
(dimensionless);
I is the maximum amplitude of the current fed to the transmitter
dipole in amperes;
T is the period of the alternating current fed to the transmitter
dipole in seconds;
r is the distance between the transmitter dipole and the receiver
dipole in meters;
.alpha. is the angle between the field detection axis through the
receiver dipole and the direction of the transmitter dipole;
and
H.sub..alpha. is the magnetic field detected by the receiver
dipole.
30. A sub-soil prospecting method according to claim 28, further
comprising the step of carrying out at the same prospecting area a
direct-current electric prospecting method comprising the steps
of:
placing in the ground two exciter electrodes at a distance from
each other;
placing two closely spaced detector electrodes in the ground
between the exciter electrodes;
applying a direct-current potential across the exciter
electrodes;
measuring the resultant potential between the detector
electrodes;
repeating said potential measurement for different distances
between the exciter electrodes; and
constructing a direct-current apparent sub-soil resistivity curve
as a function of the distance between the exciter electrodes.
31. A sub-soil prospecting method according to claim 30, further
comprising the step of combining the measurements taken using said
direct-current prospecting method with the receiver dipole output
signals of the alternating current prospecting method to construct
a relatively less ambiguous model of the sub-soil.
32. A sub-soil prospecting method according to claim 31, wherein
said step of combining includes the steps of:
determining an initial model of the subsoil;
determining theoretical direct-current and alternating-current
apparent resistivity curves corresponding to the initial model;
adjusting the theoretical curves to the constructed curves using
the least squares successive approximations technique; and
adjusting the initial model to agree with the adjusted theoretical
curves.
Description
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to sub-soil prospecting using
electro-magnetic techniques.
2. DESCRIPTION OF THE PRIOR ART
A first known sub-soil prospecting process, known as "electric
prospecting or sounding" uses direct current. Two excitation
electrodes are inserted into the earth at a distance D from each
other, and a potential difference is applied between them causing a
current to pass through the earth. Two detector electrodes spaced
by a distance d are arranged between the two excitation electrodes.
The potential difference between the detector electrodes is then
measured. A series of such measurements is taken for different
values of the distance D between the two exciter electrodes. With
the values of the distance d and the applied voltage known it is
possible to convert the voltages measured at the detector
electrodes for various distances D, into values of apparent
sub-soil resistivity associated with those values of D.
From a curve of apparent resistivities a sub-soil model can be
deduced, defined by a sequence of resistivity values associated
with depths of layers. Several processes for automatically
processing resistivity curves obtained by the above method have
already been proposed, particularly in the journal "Geophysical
Prospecting", Vol. XVIII, No. 2, June 1970, page 157: "Automatic
Processing of Electric Soundings", G. KUNETZ and J. P. ROCROI;
Geophysical Prospecting, Vol. No. 21, No. 3, September 1973, p.
543, "Example of Application of Automatic Processing to the
Interpretation of Electric Soundings", A. DUPRAT, F. GOLE and J. P.
ROCROI.
A second known process of sub-soil prospecting (called the
"induction sounding process") uses a transmitter dipole radiating
an alternating electro-magnetic field, and a receiver dipole
arranged to detect the resultant magnetic and/or electric field. By
applying MAXWELL's equations, it is theoretically possible to
deduce from the field detected by the receiver coil, and from the
distance between the two coils, information on the resistivity of
the various layers of the sub-soil. However, the solution of these
equations is extremely complex. To solve them, it is necessary to
use powerful numerical computing means and even then the computing
time required is very substantial. As a result, except in very
special cases, induction prospecting of the sub-soil has not
hitherto been the subject of extensive industrial development.
Also known is a magneto-telluric detection process in which no
artificially generated electric or magnetic fields is emitted
towards the sub-soil. The process simply involves the detection by
means of dipoles of the component of the electric field (along a
horizontal direction) and the component of the magnetic field
(along another horizontal direction, perpendicular to the
foregoing) resulting from the currents naturally circulating in the
sub-soil. Although useful information can be obtained by computer
processing of the resultant measurements the process is by its very
nature limited in usefulness because of the random existence of
telluric currents.
Moreover, if only one of the foregoing sub-soil prospecting
processes is used, for example the electric sounding process, even
after processing of the measurements there remains some
indetermination as regards the structure of the sub-soil. Indeed,
various soil structures can give the same resistivity curve for the
direct current measurements depending upon the distance set between
the exciter electrodes, and such ambiguities are enhanced by the
limited accuracy of measurements taken.
To enable a better determination of sub-soil structure, it has been
proposed to combine measurements made using the electric sounding
process and the magneto-telluric process, in such a manner as to
obtain a more precise model of the sub-soil (Geophysical
Prospecting, Vol. XXIII, No. 4, 1975, "Contribution to the Study of
Equivalence in Electric Prospection (Direct and Magneto-Telluric
Current)" J. P. ROCROI).
However, because of the very long computation times required for
the processing of induction prospecting measurements, it has been
hitherto economically impossible to combine electric and induction
prospecting to improve the sub-soil model.
It is therefore an object of the present invention to provide a
simplified process of induction sub-soil prospecting thereby
enabling the combination of direct current measurements and
alternating current induction measurements.
SUMMARY OF THE INVENTION
The present invention is based on the following observations made
by the Applicants:
(1) When use is made of a vertical magnetic field transmitter
dipole fed with an alternating current, and of a receiver dipole
orientated to receive a substantially radial horizontal magnetic
field, the radial horizontal magnetic field detected assumes
asymptotic values when the frequency of the transmitted alternating
current is decreased.
(2) These asymptotic values of the radial horizontal magnetic field
are proportional to the angular velocity of the alternating current
applied to the transmitter coil, that is, to the frequency of that
current. The asymptotic values are also inversely proportional both
to the apparent resistivity of the sub-soil to alternating current
and to the distance between the transmitter and receiver
dipoles.
(3) The apparent sub-soil resistivity to alternating current thus
obtained can be expressed simply as a function of the individual
resistivities and thicknesses of various layers of the
sub-soil.
The induction sub-soil prospecting process of the present
invention, thus comprises the steps of placing above the ground a
magnetic transmitter dipole and a magnetic receiver dipole spaced
from each other, energising the transmitter dipole with low
frequency alternating current, and measuring the receiver dipole
output to determine a low frequency asymptotic value of the
magnetic field at the receiver dipole for the particular
transmitter-receiver dipole distance used.
Preferably the transmitter dipole is vertical and the receiver
dipole is orientated to detect the horizontal magnetic field. The
detection axis of the receiver dipole is directed substantially
towards the transmitter dipole in such a manner as to detect the
magnetic radial horizontal field.
By varying the distance between the transmitter dipole and the
receiver dipole and repeating said determination of a magnetic
field low frequency asymptotic value for different
transmitter-receiver dipole distances, an apparent resistivity
curve can be obtained as a function of this distance. This curve
together with known geological data can be used to construct a
sub-soil model.
It is also advantageous to use a second receiver dipole orientated
to detect the vertical magnetic field. Measurement of the vertical
magnetic field makes it possible to determine whether an assumption
that the sub-soil is of tabular structure is roughly permissible.
Taken in combination with the radial horizontal magnetic field, it
also enables variations of the maximum amplitude of the alternating
current applied to the transmitter dipole to be ignored.
At the same time as the alternating current apparent resistivity
measurements are made using the process of the invention,
measurements can also be made of the direct current apparent
resistivity using a standard electric prospecting technique. A
comparison of these two apparent resistivities enables an improved
model of the sub-soil to be obtained.
Thus a preferred process of the invention involves first of all
constructing a model of the sub-soil from the apparent resistivity
curve obtained using a direct current electric prospecting process.
The ambiguities of this model are then reduced using the
alternating current apparent resistivity curve.
It is, of course, possible to proceed the other way around, that is
to prepare an initial sub-soil model from the alternating current
resistivity curve and then eliminate certain ambiguities using the
direct current resistivity curve.
BRIEF DESCRIPTION OF THE DRAWINGS
Sub-soil prospecting processes according to the invention will now
be particularly described, by way of example, with reference to the
accompanying diagrammatic drawings, in which:
FIG. 1 is a cross-section of a layered sub-soil showing the
arrangement above the ground surface of transmitter and receiver
dipoles used in carrying out the prospecting process of the
invention;
FIG. 2 is a graphic representation of the magnetic field as a
function of the period of transmitter radiation in logarithmic
co-ordinates, showing the asymptotic values of the magnetic field
as a function of the period; and
FIG. 3 is a graph showing a direct current resistivity curve and
three alternating current resistivity curves.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Shown in cross-section in FIG. 1 is a sub-soil composed of a number
of layers numbered from l to n. A typical layer j extends between
the depth p.sub.j-1 and depth p.sub.j. The ground surface is at
depth p.sub.o.
A transmitter dipole 1 and a receiver dipole 2 are assumed to be at
a reference depth i.e. at depth zero. Preferably the level of the
ground surface p.sub.o is very close to zero, that is, the dipoles
1 and 2 are placed practically at the level of the ground
surface.
The transmitter dipole 1 is a vertical dipole transmitting a
magnetic field and is constituted by a horizontal loop placed on
the surface of the ground, or of a ferro-magnetic core coil, whose
axis is perpendicular to the ground. The transmitter dipole 1 is
connected to a source of alternating current 10, which supplies to
it a current of maximum amplitude I, and of period T. (It is well
known of course that period T is the inverse of frequency, and that
angular velocity .omega. is equal to 2.pi./T). An efficiency
coefficient e is defined for the dipole 1 such that the magnetic
field that it transmits, expressed in ampere/meters is equal to the
product e.I.
The receiver dipole 2 comprises a coil with an air or
ferro-magnetic core. The coil is arranged in such a manner that its
axis is horizontal and passes practically through the axis of the
transmitter dipole 1. In practice, there may remain a small angle
.alpha. between the axis of the receiver dipole 2 and the direction
of the point of transmission relative to the position of the
receiver dipole. The distance between the dipoles 1 and 2 is
denoted by r. The sensitivity of the receiver dipole 2 to the
magnetic field to which it is subjected is denoted by S.
The receiver dipole 2 is connected to an amplification unit 20 with
display or recording facilities. This unit 20 receives the
alternating voltage detected by the receiver dipole 2, and
amplifies it by an amplification factor G, to produce an output
voltage U which is displayed or recorded. Thus, the magnetic field
at the receiver dipole is expressed by the following relationship:
##EQU1##
In this equation, the first term represents the magnitude of the
magnetic field H.alpha. detected at the receiver dipole, the index
.alpha. indicating that the field is not quite radial relative to
the transmitter dipole 1 but is a field along a direction making a
small angle .alpha. with the radial direction. The value H.alpha.
is placed between two vertical lines to indicate that the term
represents maximum amplitude of the magnetic field taken as an
absolute value.
Where the unit 20 has data recording facilities (for example
recording on magnetic tape) not only is the value of the voltage U
recorded but also the value of the maximum current I applied to the
transmitter dipole 1 at the time of measurement, and the frequency
or the period of this alternating current. The transmitter-receiver
distance r is also recorded. When the unit 20 simply displays the
value of the voltage U measured, a device for displaying of the
maximum current I supplied from the source 10 is also provided. The
operator also notes the frequency of the current fed to the
transmitter dipole 1 and the transmitter-receiver distance r.
FIG. 2 illustrates a typical set of measurements made on site and
clearly shows that the received magnetic field has an asymptotic
value at low frequencies. In this Figure, the absolute value of the
radial magnetic field is shown as an ordinate while the period T
expressed in seconds is shown as abscissa. The co-ordinate scales
are both logarithmic. The points 31 to 36 corresponding to on site
measurements define a curve whose concavity faces downwards.
Starting from point 37, the curve can be seen to tend towards a
straight line as the period T increases. The curve thus tends
towards an asymptote at low frequencies, that is the magnetic field
takes on asymptotic values at low frequencies.
In the asymptotic region of the curve, the field is proportional to
frequency, and thus inversely proportional to the period:
Taking into account that in FIG. 2 H and T are shown on logarithmic
scales, we have:
The graph of log .vertline.H.vertline. as a function of log T is
therefore a straight line descending at 45.degree. from the left to
the right.
Starting from these observations the Applicants endeavoured to
deduce simply from the asymptotic values of the magnetic field at
the low frequencies a value of the apparent resistivity as a
function of the distance r. Further, the Applicants also sought to
determine relationships between the apparent resistivity thus
obtained and the individual resistivities of the various layers of
the sub-soil.
These steps necessitated theoretical calculations from MAXWELL
equations as a starting point, calculations which are summarised in
the Appendix 1 to the present description, while the formulae
concerned are given in Appendix 2.
The final equation XI of Appendix 2 gives the ratio between the
radial magnetic field Hr, the apparent resistivity .rho. a and the
distance r between the transmitter dipole and the receiver dipole,
in units of the rationalised Giorgi system.
If the absolute value of the maximum field amplitude is taken, the
imaginary number i (which indicates a 90.degree. phase shift)
disappears. The factor SI which indicates the field strength of the
transmitter dipole, is now written e.I, account being taken of the
definitions made above. Moreover, since in practice measurement is
not made of the radial field, but of a field forming a small angle
.alpha. with it, it is necessary to multiply the second term of
equation XI by cos .alpha.. Finally the field H.alpha. is expressed
according to the practical unit known as gamma (or .gamma.) well
known to technicians. Equation XI thus now becomes: ##EQU2##
wherein
.rho.a is the resistivity in ohms-meter;
e is the efficiency coefficient of the transmitter dipole
(dimensionless);
I is the maximum amplitude of the current fed to the transmitter
dipole in amperes;
T is the period of the current fed to the transmitter dipole in
seconds;
r is the distance between the transmitter dipole and the receiver
dipole is meters;
.alpha. is the angle between the field detection axis through the
receiver dipole and the direction of the transmitter dipole;
and
H.alpha. is the field detected by the receiver dipole.
Moreover, inspection of equation X in Appendix 2 shows that there
is a simple arithmetic relationship (not involving integrals)
between the apparent resistivity .rho. a of the sub-soil and the
conductivities .sigma.j--or the resistivities .rho.j=1/.sigma.j of
the various layers of the sub-soil.
These relationships being known, it becomes possible to use the
apparent resistivity curve (dependent on the transmitter-receiver
distance) obtained by alternating current induction
prospecting.
The prospecting process of the invention relates to induction
prospecting based on the foregoing relationships. A practical
illustration of induction prospecting in accordance with the
invention will now be given:
(a) A magnetic field is emitted from the dipole 1 at a frequency
considered to be sufficiently low to meet the requirements of the
low frequency approximation. The alternating current I applied to
the transmitter dipole and the voltage U perceived by the receiver
dipole are measured. The magnetic field is calculated at the level
of the receiver dipole by means of equation (1) given above. The
magnetic field is expressed in gammas. The value thus obtained is
transferred to a system of logarithmic co-ordinates where the
magnetic field .vertline.H.alpha..vertline. is expressed as a
function of the period T (which is the inverse of frequency).
(b) A lower frequency is transmitted and the same operations are
repeated.
(c) A third, still lower, frequency is emitted and the same
operations are again repeated.
After that, it is easy to check from the measurements taken whether
the low frequency approximation is good. If this is the case, the
curve obtained approaches a straight line asymptote slanting at
45.degree. downwards from left to right. It is the position of this
straight line on the graph which gives the proportionality factor k
connecting the magnetic field to the inverse of the period in
equation (2) given above.
If the approximation is not verified, the measurements are repeated
on at least two frequencies of still lower values, until
satisfactory results are obtained.
To obtain immediately the apparent resistivity .rho.a, it is
possible to modify slightly equation (4) to form the term composed
of the product of the period T and the magnetic field
.vertline.H.alpha..vertline.. Since this product is equal to k,
equation (4) becomes: ##EQU3##
All the elements of this equation (5) are known, and it is
therefore easy to derive the apparent resistivity .rho.a. This
rapid derivation of the resistivity .rho.a can be seen to be due to
the fact that the factor k can be found from two measurements taken
in the field where the low frequency approximation is valid.
In a modified process according to the invention, the vertical
magnetic field H.sub.z is also measured by a second receiver dipole
placed at the same point as the first. If the sub-soil is of
tabular structure, the apparent resistivity can be expressed as a
function of the ratio of the vertical magnetic field
.vertline.H.sub.z .vertline. to the radial magnetic field
.vertline.H.sub.r .vertline. with the value of the current fed to
the transmitter dipole being eliminated: ##EQU4##
It is therefore possible either to measure simultaneously H.sub.z,
H.sub.r and I and use both equations (5) and (6) to check the
hypothesis that the sub-soil is tabular in structure, or, if the
validity of this hypothesis can be reasonably assumed, then only
H.sub.r and H.sub.z need be measured and their ratio used to
determine the apparent resistivity .rho.a from the equation
(6).
Whichever of the above processes is used, the operations involved
are repeated for different values of the distance r between the
point of transmission and the point of reception, to obtain a curve
of the resistivity .rho.a as a function of said distance r, and
this curve will hereinafter be referred to as the "alternating
current resistivity curve".
Preferably, increasing values of the distance r are used because it
has been observed that in this case the frequencies previously used
continue to verify the conditions of the low frequency
approximation.
Equations X and XI of Appendix 1 and 2 relate the alternating
current apparent resistivity .rho.a and the resistivities
1/.sigma.j of the various layers of the sub-soil, as well as their
depths p.sub.j and the distance r. It is therefore possible to
produce in usual manner a model of the sub-soil from only the
alternating current apparent resistivity curve, that is from among
the infinity of the sub-soil models likely to correspond to said
curve, one is selected which is compatible with the known
geological peculiarities of the area under study.
It has, however, been found preferable to combine with the
induction prospecting process an electric prospecting process
giving a direct current resistivity curve .rho.c as a function of
the half-length of the base line (half the distance between the two
exciter electrodes). The direct current resistivity curve .rho.c is
processed in standard manner, and the alternating current
resistivity curve .rho.a is used to reduce the indeterminate
features of the model obtained from the direct current curve, this
technique being called "reducing the limits of the equivalence"
between the model proposed and the results of the electric and
induction prospecting processes.
The basic steps of this technique are as follows:
(a) a likely model of the sub-soil is selected, defined by layer
resistivities .rho.j and heights of layers h.sub.j, with h.sub.j
=p.sub.j -p.sub.j-1 ;
(b) the corresponding theoretical curves are determined for the
direct current resistivity .rho.c as a function of the base line
half length, and for the alternating current resistivity .rho.a as
a function of the distance r between transmitter dipole and
receiver dipole;
(c) the initial model is improved by comparing the theoretical
curves .rho.c and .rho.a with the corresponding curves obtained on
site, and adjusting them according to the technique of successive
approximations, using the method of least squares, described in the
already-quoted works "Geophysical Prospecting" Vol. XVIII, No. 2,
June 1970, pp. 170-175, as well as "Geophysical Prospecting" Vol.
XXII, No. 4, 1975, pp 768-778.
Advantageously, at the start of step (c) the direct current ground
resistivity curve .rho.c and/or of the alternating current
resistivity curve .rho.a, is modified in such a manner as to
replace the curve by the closest curve corresponding to a tabular
structure of the sub-soil ("Geophysical Prospecting", Vol. XVIII,
No. 2, June 1970, pp. 187-188).
With this technique, it is obviously useful that the selection of
the initial model should be made by a geophysicist on the
examination of the curves .rho.c and .rho.a obtained. As an
alternative, it is possible to determine by calculation the model
of sub-soil corresponding, for example, to curve .rho.c ("inversion
of the resistivity curve").
Moreover, it is advantageous that step (b) above should be followed
by a determination of the limits of the equivalence between the
results .rho.c of the electric prospecting process and the results
.rho.a of the induction prospecting process. To do this, it is
possible to make use of the technique described in "Geophysical
Prospecting" Vol. XXIII, No. 4, 1975, pp. 766-770. This additional
operation permits the compatibility of the curves .rho.c and .rho.a
with each other to be estimated, for the initial model
proposed.
The technique set out above enables refined sub-soil model to be
obtained the reliability of which can be checked. Of course, if the
reliability is found to be inadequate, the entire process may be
restarted from a different initial model.
An example will now be given with reference to FIG. 3 showing how
induction prospecting can dispel ambiguities of an electric
prospecting process.
The table below gives three examples of sub-soil structure:
______________________________________ Sub-soil No. 1 Sub-soil No.
2 Sub-soil No. 3 h.sub.j .rho..sub.j h.sub.j .rho..sub.j h.sub.j
.rho..sub.j ______________________________________ 10 50 10 50 10
50 15 500 10 750 17 500 10 100 5 200 .infin. 10 .infin. 10 .infin.
10 ______________________________________
In FIG. 3 these three different sub-soils give practically the same
direct current resistivity curve (curve C). On the other hand, the
alternating current curves A1, A2 and A3 are different. In such a
case, the process described hereinabove will make it possible to
obtain a much less ambiguous sub-soil model than the electric
prospecting process alone would have done. This possibility of
reducing ambiguity is one important consequence of the induction
prospecting process of the invention.
Moreover, it has been observed that the standard direct current
prospecting techniques are unable to clearly distinguish a number
of contiguous and resistant underground layers found between a
conductive surface layer and a conductive substratum. On the other
hand, induction alternating current prospecting in accordance with
the invention is able to distinguish these layers far better (see
sub-soils Nos. 1, 2 and 3 of the above table and FIG. 3). Such a
situation is a standard one in water-bearing soils; thus processes
according to the invention are particularly useful in
hydrogeology.
In the preferred method of the invention described above the
magnetic transmitter dipole is vertical, the measured magnetic
field is horizontal and substantially radial, and the transmitter
and receiver dipoles are placed at ground level.
As already described it is clear that it is possible to detect the
horizontal magnetic field in a non-radial direction, which brings
in the angle .alpha. between the detection direction and the radial
direction (a direction passing through the transmission and
reception points)--see equation 5. Indeed, the tangential
horizontal magnetic field is in principle nil and cannot disturb
measurement. However, if the angle .alpha. assumes high values, the
term "cos .alpha." in equation 5 decreases, and with it the
sensitivity of measurement.
Where the transmitter and receiver dipoles are placed above ground,
the magnetic field is described by equation IX which has one term
more than equation X. However, the simple ratio (that is, without
integrals) between the measured values of the field and the
conductivities .sigma.j of the various layers, remains.
In Appendix 1 in fine, it will be noted that the low frequency
approximation is also valid when the magnetic transmitter dipole is
horizontal. Although the formulae involved are a little more
complex, as the electric and magnetic fields derive from two scalar
potentials, nevertheless, there still remains a simple ratio,
without integrals, between the components of the magnetic field and
the conductivities .sigma.j of the various layers.
As a result, it is also possible to carry out the prospecting
process of the invention with a magnetic transmitter dipole of the
horizontal type, and a vertical magnetic field receiver dipole, for
example.
APPENDIX 1: EXPRESSION OF THE ASYMPTOTIC VALUES OF THE MAGNETIC
FIELD PRODUCED BY A MAGNETIC DIPOLE IN THE PRESENCE OF A TABULAR
SUB-SOIL
It is current practice to assume that the sub-soil is of a tabular
structure, that is, bedded, and composed of n layers. In FIG. 1,
layer j extends from the depth z=p.sub.j-1 to the depth z=p.sub.j ;
its electric conductivity is entered as .sigma..sub.j. The surface
of the soil is thus at level z=p.sub.o .gtoreq.0.
The magnetic transmitter dipole is in air, at level z=0. Through it
passes an alternating current of maximum amplitude 1 and of angular
velocity .omega.. The surface of the dipole, denoted S, is equal to
the elementary surface of a turn, multiplied by the number of
turns.
It is necessary to solve the MAXWELL equations to determine the
electric field and the magnetic field produced by the dipole. The
work "Mining Geophysics", Vol. 2, 1967, published by the Society of
Exploration Geophysicists, gives on page 50 a solution of the
MAXWELL equations, with the usual approximations on the
characteristics of the sub-soil; no electric charge or magnetic
hysteresis in the sub-soil, negligible influence of the electric
induction vector D, proportionality of this same vector D to the
electric field vector E, and finally proportionality of the current
density vector .delta. to the electric field, i.e. .delta.=.sigma.,
E.
Under these conditions, the electric field vector E and the
magnetic field vector H when expressed in cylindrical co-ordinates
(r.psi.z) can be written as linear combinations of two particular
solutions. One of these two particular solutions corresponds to a
magnetic field vector whose component along the z direction is nil.
The other solution corresponds to an electric field whose component
along the z direction is nil. With each of these two particular
solutions it is possible to associate a respective scalar potential
from which the electric field and the magnetic field are
derived.
The Applicants first of all noticed that, for a vertical magnetic
dipole, the vertical component (along the z axis) of the electric
field is nil, which corresponds to the second particular solution.
Under these conditions, the electric and magnetic fields henceforth
depend only upon this second particular solution, and their
components r, .psi. and z are expressed as a function of the scalar
potential V associated according to the equations (I) (all the
equations are given in Appendix 2).
The scalar potential V from which electric field vectors E and
magnetic field vectors H are defined must for its part satisfy the
conditions set forth in equation II, a and b.
In equations (I) and (II) .mu. defines the magnetic permeability of
the medium concerned, and i denotes the imaginary unit well known
in complex numbers. As the components of the vectors E and H, as
well as the scalar potential V, are expressed in the form of a sine
function of time, their time derivative is obtained by multiplying
them simply by -i.omega..
Bearing in mind the foregoing notations, equations (I) and (II)
correspond to those given in the referenced work ("Mining
Geophysicists"). Similar equations are also found, expressed in a
more general system of co-ordinates, in the work entitled
"Complements de Mathematiques" by Andre Angot, 5th ed. 1965,
Collection Technique et Scientifique du CNET, p. 335; introduction
of the specific conditions of the present application into these
equations will result in equations identical to equations (I) and
(II).
It is known that equation (IIa) can be integrated by using Bessel
functions. The solution of equation (IIa) in air gives a primary
potential V.sub.o expressed by equation III.
In this equation, function J.sub.o is a Bessel function of the
first type and of order 0, .lambda. is an integration variable, and
d is distance .sqroot.r.sup.2 +z.sup.2.
From this can be derived the expression for the primary fields
transmitted directly through the air between the transmitter dipole
and the receiver dipole. For example, the radial component of the
primary magnetic field in the air is given by equation IV.
The word primary associated with the electric and magnetic fields
and with the potential from which they derive, indicates that these
are fields produced directly in the air by the transmitter dipole.
Added to these are secondary field which are due to the presence of
the sub-soil and which are still to be determined.
To determine these secondary fields, the tangential component of
the electric field in each of the layers is first derived. For the
layer j this field is given by equation V.
In this equation, A.sub.j corresponds to a wave propagating
downwards and B.sub.j to a wave propagating upwards. Function
J.sub.1 is a Bessel function of the first type and of order 1.
The tangential component of the magnetic field is expressed by an
equation of similar form.
The Applicants observed that it is not necessary to revert to the
scalar potential V.sub.j in the various layers to express the
continuity of the fields at the level of the surfaces of separation
of the various layers. It is sufficient to consider the continuity
of the tangential component of the electric field and that of the
magnetic field. Recurrence relationships are then obtained between
parameters A.sub.j, B.sub.j and R.sub.j (see equations VI).
In air, A.sub.0 =0, and in the last layer, B.sub.n+1 =0. After
having solved the recurrence relationships, it is possible to
derive the expression for the secondary magnetic field which is
produced in the air due to the various layers of the sub-soil. The
component H.sub.r of this secondary magnetic field in the air is
given by equation (VII).
The integral contained in this equation gives rise to fairly
lengthy and intricate calculations.
It is at this stage that low frequency approximations can be
usefully introduced to greatly simplify the recurrence relationship
between the above-mentioned coefficients A.sub.j and B.sub.j. With
such approximations the Applicants noted that factor B.sub.o of
equation (VI) can be expressed as a linear combination of the
conductivities .sigma..sub.j of the various layers.
Thus, for the radial component H.sub.r of the secondary magnetic
field the relationship given in equation VIII is obtained.
The total radial horizontal magnetic field at the level of the
receiver dipole is given by the sum of the primary value (equation
IV) and of the secondary value (equation VIII)--see equation
(IX).
If we assume that the transmitter and receiver dipoles are at
ground level, we have P.sub.o =0 and z=0, which simplifies equation
(IX) to the form shown in equation X.
Whereas equation (VII) contains an integral whose numerical
evaluation is intricate, the equations (IX) and (X) enable an
asymptotic value of the radial magnetic field to be obtained which
is simply a summation of terms depending upon the conductivities
and depths of the various layers.
If the expression between the square brackets in equation (X) is
denoted by 1/.rho..sub.a, where .rho..sub.a corresponds to the
apparent resistivity of the sub-soil at a point of radius r,
equation (X) assumes the form given in equation XI.
It will now be clearly seen that the radial horizontal magnetic
field assumes linear asymptotic values as a function of frequency,
or of the angular velocity .omega. appearing in equation (XI).
Moreover, these asymptotic values are inversely proportional to the
apparent resistivity .rho..sub.a of the sub-soil, as well as to the
distance r relative to the transmitter dipole.
The low frequency approximation also gives greatly simplified
expressions for both the vertical magnetic field Hz, and the
electric field.
The foregoing results have been derived for the case where the
transmitter dipole is vertical. Where the transmitter dipole is
horizontal, the expressions of the magnetic and electric fields in
the air depend, not upon a single scalar potential as was the case
previously, but upon two scalar potentials of the general case.
Nevertheless, the use of low frequency approximations still give
rise to greatly simplified expressions for the magnetic field at
the level of the receiver dipole. Although these expressions are
less simple than the equations (IX) to (XI), they also have the
advantage that the fields are expressed as a summation of terms
related to the conductivities of the various layers, instead of
comprising the integral of a Bessel function. The numerical
evaluation of these expressions is therefore also considerably
facilitated.
APPENDIX 2: EQUATIONS (I) to (XI) ##EQU5##
* * * * *